Does anybody have thoughts on why there are such different strategies employed at different times to refer to oneself?

On the one hand, there is the uneqivocable "ἐγὼ Παῦλος", used 6 times in his epistles, and the other hand there is the inferential style of ὁ μαθητὴς ὃν ἐφίλει ὁ Ἰησοῦς, used by the Evangelist when speaking of himself.

Using the ἐκεῖ as either a marker of distance from the speaker, so either distain, or from a visual/ perspective point of view, the person would appear to be of smaller stature. When used in syntax, the meaning might only be demonstrative, but in this case (and in Matthew 27:63 ἐκεῖνος ὁ πλάνος), the meaning seems more connative than grammatical. If that is the case, the circumloqacious self-referentiality may be being accompanied by a form of self-deprecation or modesty - the one out of the lime-light, "that unimportant man who was loved by Jesus". It may also refer to a person who is respectfully (as opposed to disrespectfully) removed from the present company.

Just to point out how that thinking differs from BDAG (esp. a. γ.), let me say that BDAG is compiled on the exclusive basis that ἐκεῖνος is a relative pronoun (in all cases). Referring in a.γ, as they say, to unmentioned, but the well-known or notorious - it is a relative pronoun of what was not just mentioned. I am suggesting that the distance value of ἐκεῖνος (a lexical meaning) is sometimes used exclusively, in cases where the demonstrative (a syntactic value) is not employed. Without a syntactic role, if it needed a label, ἐκεῖνος would be an adjective.